Ballantine's theorem on Sp(2n, C)

• Main Author: Granario, Daryl Q.
• Format: text
• Published: Animo Repository 2012
• Subjects: Positive-definite functions; Symplectic groups; Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/faculty_research/11356
• Institution: De La Salle University

Among normal matrices, positive definite matrices or positive semidefinite matrices are important. Any product of two positive definite (semidefinite) matrices is diagonalizable and has positive (nonnegative) eigenvalues so such products do not fill up Mn(C) or GL(n, C), the group of n × n nonsingular matrices.. Ballantine [1, 2, 3, 4] showed that every matrix with positive de- terminant is a product of five positive definite matrices. Radjavi [11] showed that every matrix A with real determinant is a product of at most four Her- mitian matrices, and there are matrices which are not product of less. Wu [14] showed that every matrix with nonnegative determinant is a product of five positive semidefinite matrices. In the same paper, Wu also gave a char- acterization of matrices that can be expressed as a product of four positive semidefinite matrices. A recent work of Cui, Li, and Sze [5] gives a character- ization of products of three positive semidefinite matrices. We are interested in the following problem. Problem. Let G be a subgroup of GL(n, C) consisting of matrices with positive determinant. Can the matrices in G be written as a product of a finite number of positive definite matrices in G? If so, how many factors do we need to express all elements of G as products of positive definite matrices in G? In this talk, we consider the complex symplectic group Sp(2n, C): Sp(2n, C) = {A ∈ GL(2n, C) : ATJnA = Jn} , where Jn = [0 In] [−In 0] The symplectic group is a classical group defined as the set of linear trans- formations of a 2n-dimensional vector space over C which preserve the non- degenerate skew-symmetric bilinear form which is defined by Jn. It is a non-compact, simply connected, and simple Lie group [8]. Every symplectic matrix A ∈ Sp(2n, C) is nonsingular with the inverse A−1 = J −1 n A>Jn. It is obvious to see that det A = ±1 but less obvious to narrow it down to det A = 1 though it is true. So one may write Sp(2n, C) = {A ∈ SL(2n, C) : J −1 n A >Jn = A −1 }. One can see that Sp(2n, C) is invariant under ∗, the conjugate transpose. We drop the subscript if the size of the matrix Jn is clear from context. For A ∈ GL(2n, C), we define AJ = J−1ATJ, the J-adjoint of A. In block form, one can write Sp(2n, C) = {A B C D ∈ SL(2n, C) : A TC = C TA, B TD = DTB, ATD − CTB = In} .